# why does unhinged body rotate about centre of mass when off-centre force is applied? [duplicate]

Any force applied to a body creates a torque around some reference point, but if we ignore all other forces (gravity etc.) the body will rotate if and only if the torque is not zero relative to the centre of mass? How can we prove that? Why is it the centre of mass and not some other point?

If the torque around the CoM is zero, I will still be able to find infinitely many other points such that torque around all of them is not zero, but the body won't rotate at all.

• With your second point, can you give an example
– user86425
Commented Mar 26, 2017 at 18:04
• @rpfphysics A car traveling in a corner. There's a centripetal force from tyres on both axles pushing the car towards the corner. Assuming the car is traveling a constant radius corner at a constant speed, the torque around CoM is zero (it doesn't rotate faster and faster). But if you pick a point on the rear axle, the net torque relative to that point will not be zero (because you have the force from tyres on the front axle). Commented Mar 26, 2017 at 18:46
• The point which you choose must be inertial for the calculation of torque to work properly (ignoring external forces). The only solution to this point is the CoM
– user86425
Commented Mar 26, 2017 at 18:51
• @sammygerbil ok, the second link explains why a body rotates around the CoM if there's no external forces acting on it. My question is why does it rotate around CoM if we apply a force to a stationary, non-rotating rigid body. Is it because if it didn't rotate around CoM if we removed that force, it would violate the statement from the second link? Commented Mar 26, 2017 at 18:54
• Commented Mar 26, 2017 at 22:45

You can think of the center of mass as a point around which the mass distribution of the object is more or less the same. So if you kept the object on a weight balance with the tip of the weight balance at this point the object would be perfectly balanced. Now consider a uniform solid sphere, whose center of mass is at the geometric centre. Now roll it, you will see that all the points except the centre seem to move in concentric circles with a common axis, while the centre just moves in a straight line path. Since it has no rotational motion it is the best reference frame to understand the rotational motion of the other particles.

To answer this question first you must consider how is linear and angular momentum defined. (Here point C designates the center of mass).

1. Linear momentum is the product of the scalar mass with the linear velocity of a rigid body at the center of mass $$\mathbf{p} = m \mathbf{v}_C$$
2. Angualar momentum at the center of mass is defined as the product of the mass moment of inertia tensor at the center of mass and the rotational velocity vector $$\mathbf{L}_C = \mathrm{I}_C \boldsymbol{\omega}$$

The reason for this distinction is the linear momentum describes the motion of the center of mass and angular momentum the motion about the center of mass. This separation comes naturally out of the equations when summing up the movement of all the particles on a rigid body.

Once you have accepted the above definitions you find the following:

1. Net force equals the time derivative of linear momentum $$\sum \mathbf{F} = \frac{{\rm d}}{{\rm d}t} \mathbf{p} = m \frac{{\rm d}}{{\rm d}t} \mathbf{v}_C = m \, \mathbf{a}_C$$
2. Net torque about the center of mass is the time derivative of angular momentum about the center of mass $$\sum \boldsymbol{\tau} = \frac{{\rm d}}{{\rm d}t} \mathbf{L}_C = \mathrm{I}_C \left( \frac{{\rm d}}{{\rm d}t} \boldsymbol{\omega} \right) +\left( \frac{{\rm d}}{{\rm d}t} \mathrm{I}_C \right) \boldsymbol{\omega} = \mathrm{I}_C \boldsymbol{\alpha} + \boldsymbol{\omega} \times \mathrm{I}_C \boldsymbol{\omega}$$

So the key here is that a net zero force will cause the center of mass no acceleration (constant velocity = Newton's 1st law) and thus a pure torque will rotate a body about the center of mass. The differentiation of angular momentum retains the disclaimer "about the center of mass" that we establised in the first part of this answer.

The following two statements are true and dual to each other (if one is true, so must the other). They are both a result of the definition of the center of mass.

• A net pure torque accelerates a rigid body about the center of mass
• A force through the center of mass accelerates a body with pure translation.

Reference: Derivation of Newton-Euler equations of motion.

• Ok, makes sense. But lets look at another case. Lets say I have a rod in space (ignore all forces, such as gravity). If I apply a force in the point of CoM, it wont rotate, but it will create a torque about one of its end. So now I have a non zero torque about one of its ends, but the rod doesnt rotate. Why? Isn't it that a nonzero torque should creare rotational acceleration no matter what point we pick? Commented Mar 27, 2017 at 5:55
• @user5539357 because it is the net torque about the center of mass that counts only. Read my linked answer carefully where I have qualified at which point things are considered. The geometry of the system (location of forces and motions) is just as important as the magnitudes of the values involved. Commented Mar 27, 2017 at 12:48
• But what about rolling friction? Assuming no slip, the net force on the roller is $\vec{F_{net}}=\vec{F_{some \: force}}+\vec{f_{static}}$ and it is exerted on the edge of the roller. Commented Jun 21, 2020 at 1:47
• For steady rolling, then $\vec{F}_{\rm net}=0$ and the center of mass moves in a constant fashion since $\vec{a}_C=0$. Only when $\vec{F}_{\rm net} \neq 0$ the center of mass will accelerate or curve around. Commented Jun 21, 2020 at 13:03

It's just a useful convention. You can always decompose the motion of a rigid body into a translation and a rotation about some point. If you choose the point to be the CM, things are more convenient, e.g. the kinetic energy separates nearly into linear and rotational parts.

For example, one could say that a ball falling down is actually rotating about a very faraway point. It's just not very useful so we don't define "rotating" in this way.

• As a counter example suppose a rod is in space. You apply a force at one end. It will rotate and translate. You will say we can choose any point in the rod about which we can describe a rotation. Now say you applied the force at the centre of mass of the rod. It will not rotate just translate. WHY it doesn't rotate in this case ? Or can you describe a rotation about some point either inside or outside the rod ? Could you please help me in this regard ? because I know we can choose any rotation axis. But in this case the rod doesn't " look to be rotating at all" just like your example of ball Commented Jun 28, 2020 at 17:25
• Can you let me this point because I won't be able to ask a new question citing just this specific point.... Commented Jul 1, 2020 at 19:25
• @knzhou Can you provide a reference? Commented Sep 3, 2022 at 13:13